The normal curve

• Some populations match known distributions.
• Consider the experiment of rolling a die over and over.
  • This would result in a uniform distribution.
  • Each outcome is equally likely.
  • If you are told that a population is uniformly distributed you can draw inferences about that population.
• Frequently grades in a low level class have a bimodal distribution. It is not strange for a class to have grades: A 5 B 15 C 4 D 5 F 11
• Upper level classes frequently has a J shaped distribution: A 6 B 4 C 2 D 1 F 0
• Or even a skewed distribution. (A 2, B 7, C 4, D 2, F 1)
• Much of the natural world, however follows a Gaussian or Normal distribution.
• What is the distribution of sums of tossing two die?

• Sum	Frequency
  2	1
  2	2
  4	3
  5	4
  6	5
  7	6
  8	5
  9	4
  10	3
  11	2
  12	1

• This is called the normal curve.
• Due to the shape, this is called the bell curve
• Many things are assumed to be normal
  • Sizes of items (height, weight, IQ, ...)
  • Demand for items
  • Many physical phenomena
• In a normal distribution the mean, median and mode are all the same.
• The Empirical Rule states that in a population which is normally distributed, with a mean of μ and a standard deviation of σ
  • 68% of the data is within one standard deviation of the mean.
  • 95% of the data is within two standard deviations of the mean.
  • 99.7% of the data is within three standard deviations of the mean.
• Do problems 51 and 52 page 895
• Try problem 53
• The tables on page 888 889 give us the percent of the values to the left of a given score.
• μ=0, σ =1
• Check out the values.
• When we have a population with a different value of μ and σ we compute the z-score for some data.
• z = (x - μ)/σ
• Find z scores for 51 through 70.
• Using the z score, we can find the area under the curve using the tables.
• Do 38 - 48.
• Using the area, we can do other problems. 51-82)